Note first that in this proof all elements in discussed
Conditions C1) through C13) are replaced without further notice
by the corresponding elements that are used in Conditions
C3.1), C3.1.1), C10.1), C12.1) and relations (REF )
through (REF ). For example, in Condition
C3) \(B_n\)  is replaced by \(B_n^*\) .
By the assumptions about the functional spaces and operators
introduced above, Conditions C2), C4), C6), C6a),
C6b), C11), and C11a)
of Section REF  are satisfied. Indeed,
Conditions C2), C6), and C11) are satisfied by the direct
definitions of the operators whose existence is required
in these conditions.
In addition,
Conditions C4), C6a), C6b), and C11a) are satisfied
by Proposition REF .
Next, Conditions C3.1), C10.1), and C12.1) of this section
are equivalent to Conditions C3), C10), and C12) of Section
REF , respectively.
Therefore, statements (i), (iii), and (iv) of Corollary REF 
follow immediately
from statements (i), (ii), and (iii) of
Corollary REF , respectively.
To prove statement (ii), we recall that
the norm \(\Vert \cdot \Vert _{F^{(1)}({\mathbb {R}}^m)}\)  is monotone and
\({\omega }\) -continuous. So for
any \(f\in B\setminus \lbrace 0\rbrace \) , we obtain by
Proposition REF  and relation (REF )
of Condition C3.1.1)
\(&&\Vert D(f)\Vert _{F^{(1)}({\mathbb {R}}^m)}=\lim _{n\rightarrow {\infty }}\Vert D(f)\Vert _{F^{(1)}({\Omega }_n^*)}\nonumber \\&&\le \liminf _{n\rightarrow {\infty }}\left(\Vert D(f)-D_n(f_n)\Vert _{F^{(1)}({\Omega }_n^*)}^{{\beta }_1}+\Vert D_n(f_n)\Vert _{F^{(1)}({\Omega }_n^*)}^{{\beta }_1}\right)^{1/{\beta }_1}\nonumber \\&&=\liminf _{n\rightarrow {\infty }}\left\Vert D_n(f_n)\right\Vert _{F^{(1)}({\Omega }_n^*)}\le \liminf _{n\rightarrow {\infty }}\left\Vert D_n(f_n)\right\Vert _{F^{(1)}({\Omega }_n)}\nonumber \\&&\le \liminf _{n\rightarrow {\infty }}\sup _{f\in B_n\setminus \lbrace 0\rbrace }\frac{\Vert D_n(f)\Vert _{F^{(1)}({\Omega }_n)}}{\Vert f\Vert _{F^{(2)}({\Omega }_n)}}\limsup _{n\rightarrow {\infty }}\left(\Vert f_n\Vert _{F^{(2)}({\Omega }_n)} \right).\) 
Then (REF ) follows from (REF ) and (REF ).
Finally, statement (v) follows from (i), (iii), and (iv),
and statement (vi) follows from (ii), (iii), and (iv).
\(\Box \)

Examples.
There is a number of publications
[1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]} about special cases of Corollary
REF ;
here, we call them Bernstein-Nikolskii type
inequalities or inequalities of different metrics
(see also Section  A).
Examples REF , REF , and REF  below
discuss inequalities (REF ),
(REF ) and (REF )
proved in these papers in various situations.
Example 7.2 
(Multivariate Bernstein-Nikolskii type
inequalities).
Let
\(F^{(1)}({\mathbb {R}}^m)=L_q({\mathbb {R}}^m),\,{\beta }_1=\min \lbrace 1,q\rbrace ,\,F^{(2)}({\mathbb {R}}^m)=L_p({\mathbb {R}}^m),\,{\beta }_2=\min \lbrace 1,p\rbrace ,\,B=B_V\cap L_p({\mathbb {R}}^m),\,0<p\le q\le {\infty },\,{\Omega }_n=Q_{\pi n}^m,\,{\Omega }_n^*=Q_{n^{\delta }}^m,\, {\delta }={\delta }(p,m)\in (0,1),\,B_n={\mathcal {T}}_{n,V}\left|_{\Omega _n}\right.={\mathcal {T}}_{n,V},\,B_n^*={\mathcal {T}}_{n,V}\left|_{\Omega _n^*}\right.={\mathcal {T}}_{n,V},\,n\in {\mathbb {N}}.\) 
Next, let
\(d_N=\sum _{\vert {\alpha }\vert =N}b_{\alpha }D^{\alpha }\) ,
where
\(b_{\alpha }\in {\mathbb {C}}^1\)  (\({\alpha }\in {\mathbb {Z}}^m_+,\,\vert {\alpha }\vert =N\) ), and \(N\in {\mathbb {Z}}^1_+\) ,
are independent of
\(n\) , and let
\(D=d_N\left|_{{\mathbb {R}}^m}\right.=d_N,\,D_n=d_N\left|_{\Omega _n}\right.=d_N,\,D_n^*=d_N\left|_{\Omega _n^*}\right.=d_N,\,n\in {\mathbb {N}}\) .
Inequality (REF ) for \(0<p\le q\le {\infty }\)  and
equality (REF ) for \(0<p\le {\infty }\)  and \(q={\infty }\) 
were proved by the author
[8]} in the following form:
\(&& \sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_q({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}\le \liminf _{n\rightarrow {\infty }}\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_q(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}, \\&&\sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_{\infty }({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}= \lim _{n\rightarrow {\infty }}\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}. \) 
The following asymptotic results
(see [8]})
are easy corollaries of (REF ) and
() (\(0<p\le q\le {\infty }\) ):
\(&& \sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_q({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}\le \liminf _{n\rightarrow {\infty }} n^{-N-m/p+m/q}\sup _{Q\in {\mathcal {T}}_{nV}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(Q)\Vert _{L_q(Q_\pi ^m)}}{\Vert Q\Vert _{L_p(Q_\pi ^m)}}, \\&&\sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_{\infty }({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}= \lim _{n\rightarrow {\infty }} n^{-N-m/p}\sup _{Q\in {\mathcal {T}}_{nV}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(Q)\Vert _{L_{\infty }(Q_\pi ^m)}}{\Vert Q\Vert _{L_p(Q_\pi ^m)}}. \) 
For \(m=1\)  and \(V=[-{\sigma },{\sigma }]\)  relations
(REF ) and () were
established by the author and Tikhonov
[6]}.
Similar relations for \(m=1,\,V=[-{\sigma },{\sigma }]\) , and
\(N=0\)  were obtained earlier by
Levin and Lubinsky [4]},
[5]}.
A special case of \(m=1,\,V=[-1,1],\,q={\infty },\,p=1\) , and
\(N=0\)  was discussed by Taikov
[1]}, [2]}
and Gorbachev [3]}.
More references can be found in
[6]}, [8]}.
Condition C3.1.1) of approximation of entire
functions of exponential type \(V\) 
by trigonometric polynomials was verified
by using univariate and multivariate Levitan's polynomials
(see [6]} for \(m=1\)  and
[8]} for \(m>1\) ).
In addition, we choose
\(B_n^\prime &=&\Bigg {\lbrace }f_n\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace :\Bigg {(}\Vert d_N(f_n)\Vert _{L_{\infty }(Q_{\pi n}^m)}=\vert d_N(f_n)(0)\vert =1\Bigg {)}\nonumber \\&\wedge &\left.\left(\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}=\Vert f_n\Vert _{L_p(Q_{\pi n}^m)}^{-1}\right)\right\rbrace ,\qquad n\in {\mathbb {N}}.\) 
Since the set \({\mathcal {T}}_{n,V}\)  consists of \(2\pi n\) -periodic
trigonometric polynomials, we have
\(\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}=\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\vert d_N(T)(0)\vert }{\Vert T\Vert _{L_p(Q_{\pi n}^m)}},\qquad n\in {\mathbb {N}}.\) 
Therefore, by (REF ) and by the standard compactness argument
(see the proof of Theorem 1.3 in [8]} for details),
we conclude that
\(B_n^\prime \ne \emptyset \)  and in addition,
\(\sup _{T\in {\mathcal {T}}_{n,V}\setminus \lbrace 0\rbrace }\frac{\Vert d_N(T)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert T\Vert _{L_p(Q_{\pi n}^m)}}=\sup _{f_n\in B_n^\prime }\frac{\Vert d_N(f_n)\Vert _{L_{\infty }(Q_{\pi n}^m)}}{\Vert f_n\Vert _{L_p(Q_{\pi n}^m)}},\qquad n\in {\mathbb {N}}.\) 
Thus Condition C9) is satisfied for the chosen \(B_n^\prime ,\,n\in {\mathbb {N}}\) .
Next, equality (REF ) of C12.1)
holds true since for any \(f_n\in B_n^\prime \)  we have
(\(n\in {\mathbb {N}},\,s\in {\mathbb {N}}\) )
\(\left\Vert D_{n}(f_{n})\right\Vert _{F^{(1)}\left({\Omega }_{n}\right)}=\Vert d_N(f_n)\Vert _{L_{\infty }(Q_{\pi n}^m)}=\left|d_N(f_n)(0)\right|=\Vert d_N(f_n)\Vert _{L_{\infty }(Q_{\pi s}^m)}=\left\Vert D_{n}(f_{n})\right\Vert _{F^{(1)}\left({\Omega }_{s}\right)}.\) 
Further, the compactness condition of \(\lbrace B_n^\prime \rbrace _{n=1}^{\infty }\) ,
stated in (REF ) and () of C12.1)
(with a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  that is independent of \(s\) ),
is satisfied as well.
Indeed, let \(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) . Then for \(p\in (0,{\infty })\)
\(&&\sup _{n\in {\mathbb {N}}}\Vert f_n\Vert _{L_{\infty }(Q_{\pi n}^m)}\le \sup _{n\in {\mathbb {N}}}(d_0,{\mathcal {T}}_{n,V},L_{\infty }(Q_{\pi n}^m),L_p(Q_{\pi n}^m)\Vert f_n\Vert _{L_p(Q_{\pi n}^m)}\\&&= \sup _{n\in {\mathbb {N}}}(d_0,{\mathcal {T}}_{n,V},L_{\infty }(Q_{\pi n}^m),L_p(Q_{\pi n}^m)/ (d_N,{\mathcal {T}}_{n,V},L_{\infty }(Q_{\pi n}^m),L_p(Q_{\pi n}^m)=C_8<{\infty },\) 
by the choice of \(B_n^\prime \)  and
by crude estimates of the constants in multivariate
Bernstein-Nikolskii type inequalities
(see [8]}).
Therefore,
\(\cup _{n=1}^{\infty }B_n^\prime &\subseteq &\lbrace T\in \cup _{n=1}^{\infty }{\mathcal {T}}_{n,V}:\Vert T\Vert _{L_{\infty }(Q_{\pi n}^m)}\le C_8\rbrace \nonumber \\&\subseteq &\lbrace f\in B_V:\Vert f\Vert _{L_{\infty }({\mathbb {R}}^m)}\le C_8\rbrace ,\) 
since \({\mathcal {T}}_{n,V}\subseteq B_V,\,n\in {\mathbb {N}}\) ,
and the compactness property of
the right-hand side set in (REF ) is proved
in [8]}
(a special case is discussed in
[26]}).
Note also that by [8]},
\((d_N,{\mathcal {T}}_{n,V},L_{\infty }(Q_{\pi n}^m),L_p(Q_{\pi n}^m)<{\infty },\qquad n\in {\mathbb {N}}.\) 
So (REF ) for \(0<p\le q\le {\infty }\)  and
() for \(0<p\le {\infty }\)  and \(q={\infty }\)  follow from
statements (i) and (vi) of
Corollary REF , respectively.

Example 7.3 
(Multivariate
inequalities of different metrics).
Let
\(F^{(1)}({\mathbb {R}}^m)=C_0({\mathbb {R}}^m),\,{\beta }_1=1,\,F^{(2)}({\mathbb {R}}^m)=L_p({\mathbb {R}}^m),\,{\beta }_2=\min \lbrace 1,p\rbrace ,\,B=B_V\cap L_p({\mathbb {R}}^m),\,0<p\le {\infty },\,{\Omega }_n=nV^*,\,{\Omega }_n^*=\tau nV^*,\,\tau \in (0,1),\,B_n={\mathcal {P}}_{n,m}\left|_{\Omega _n}\right.={\mathcal {P}}_{n,m},\,B_n^*={\mathcal {P}}_{n,m}\left|_{\Omega _n^*}\right.={\mathcal {P}}_{n,m}\) .
Next, let
\(d_N=\sum _{\vert {\alpha }\vert =N}b_{\alpha }D^{\alpha }\) ,
where
\(b_{\alpha }\in {\mathbb {C}}^1\)  (\({\alpha }\in {\mathbb {Z}}^m_+,\,\vert {\alpha }\vert =N\) )
and \(N\in {\mathbb {Z}}^1_+\) , are independent of
\(n\) , and let
\(D=d_N\left|_{{\mathbb {R}}^m}\right.=d_N,\,D_n=d_N\left|_{\Omega _n}\right.=d_N,\,D_n^*=d_N\left|_{\Omega _n^*}\right.=d_N,\,n\in {\mathbb {N}}\) .
Inequalities (REF ) were proved by the author
[10]} in the following form:
\(&&\limsup _{n\rightarrow {\infty }}\sup _{P\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(P)(0)\vert }{\Vert P\Vert _{L_p(nV^*)}}\le \sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\vert d_N(f)(0)\vert }{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}\nonumber \\&&= \sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_{\infty }({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}\le \liminf _{n\rightarrow {\infty }}\sup _{P\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(P)(0)\vert }{\Vert P\Vert _{L_p(\tau nV^*)}}\) 
for \(\tau \in (0,1)\) .
By letting \(\tau \rightarrow 1-\)  in (REF ),
we arrive at the following asymptotic result
for \(0<p\le {\infty }\) 
(see [10]}):
\(\sup _{f\in (B_{V}\cap L_p({\mathbb {R}}^m))\setminus \lbrace 0\rbrace }\frac{\Vert d_N(f)\Vert _{L_{\infty }({\mathbb {R}}^m)}}{\Vert f\Vert _{L_p({\mathbb {R}}^m)}}= \lim _{n\rightarrow {\infty }} n^{-N-m/p}\sup _{Q\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(Q)(0)\vert }{\Vert Q\Vert _{L_p(V^*)}}.\) 
For \(m=1,\,V=[-1,1]\) , \(V^*=[-1,1]\) ,
and \(d_N(f)=d^Nf/dx^N\) ,
equality (REF ) was established in
[7]}. In case when
\(V\)  and \(V^*\)  are the unit ball
\({\mathfrak {B}}_1^m\)  and \(d_{2N}=\Delta ^N\)  is the polyharmonic operator,
(REF ) was proved in [9]}
by reducing the multivariate problem to a univariate one
in weighted spaces (see Example REF ).
Condition C3.1) of approximation of entire
functions from \(B_V\cap L_p({\mathbb {R}}^m)\) 
by algebraic polynomials was verified in
[32]} and [10]}
by using approximation methods developed by Bernstein
[34]} (see also [35]}
and [36]}) and by the author
[32]}, [10]}.
In addition, we choose
\(B_n^\prime =\left\lbrace f_n\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace :\left(\vert d_N(f_n)(0)\vert =1\right)\wedge \left(\sup _{P\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(P)(0)\vert }{\Vert P\Vert _{L_p(nV^*)}}=\Vert f_n\Vert _{L_p(nV^*)}^{-1}\right)\right\rbrace ,\,\, n\in {\mathbb {N}}.\) 
By the standard compactness argument
(see the proof of Theorem 1.2 in [10]} for details),
we conclude that
\(B_n^\prime \ne \emptyset \)  and in addition,
\(\sup _{P\in {\mathcal {P}}_{n,m}\setminus \lbrace 0\rbrace }\frac{\vert d_N(P)(0)\vert }{\Vert P\Vert _{L_p(nV^*)}}=\sup _{f_n\in B_n^\prime }\frac{\vert d_N(f_n)(0)\vert }{\Vert f_n\Vert _{L_p(nV^*)}},\qquad n\in {\mathbb {N}}.\) 
Thus Condition C9) is satisfied for the
chosen \(B_n^\prime ,\,n\in {\mathbb {N}}\) .
Next, it is easy to verify that equality (REF ) of C12.1)
holds true in this case as well
since for any \(f_n \in B_n^\prime \) 
we have
\(\left\Vert D_{n}(f_{n})\right\Vert _{F^{(1)}\left({\Omega }_{n}\right)}=\left\Vert d_N(f_n)\right\Vert _{C_0\left(nV^*\right)}=\left\Vert d_N(f_n)\right\Vert _{C_0\left(sV^*\right)}=\left\Vert D_{n}(f_{n})\right\Vert _{F^{(1)}\left({\Omega }_{s}\right)},\quad n\in {\mathbb {N}},\quad s\in {\mathbb {N}}.\) 
Further, the compactness condition of
\(\lbrace B_n^\prime \rbrace _{n=1}^{\infty }\) , stated in
(REF ) and () of C12.1)
(with a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\) 
that is independent of \(s\) ),
is satisfied. The proof of this fact given in
[10]} is based on results
of Bernstein [34]} (for \(m=1\) )
and the author [10]}
(for \(m>1\) ).
Note also that by a crude estimate of
the constant in a multivariate inequality
of different metrics
(see [10]}),
\((d_N,{\mathcal {P}}_{n,m},C_0(nV^*),L_p(nV^*)<{\infty }\)  for \(n\in {\mathbb {N}}\) .
So (REF ) for \(p\in (0,{\infty }]\)  follows from
Corollary REF  (v).

Example 7.4 
(Univariate
inequalities of different weighted metrics).
Let \(d\mu (x)=\vert x\vert ^{2\nu +1}dx,\,\nu \ge -1/2\) ,
be the univariate weighted measure on \({\mathbb {R}}^1\) . Next, let
\(F^{(1)}({\mathbb {R}}^1)=C_0({\mathbb {R}}^1),\,{\beta }_1=1,\,F^{(2)}({\mathbb {R}}^1)=L_{p,\mu }({\mathbb {R}}^1),\,{\beta }_2=\min \lbrace 1,p\rbrace \) ,
\(B=B_{1,e}\cap L_{p,\mu }({\mathbb {R}}^1):=\lbrace f\in B_{[-1,1]}:f(-x)=f(x),\,x\in {\mathbb {R}}^1\rbrace \cap L_{p,\mu }({\mathbb {R}}^1),\) 
\(0<p\le {\infty },\,{\Omega }_n=[-n,n],\,{\Omega }_n^*=[-\tau n,\tau n],\,\tau \in (0,1),\,n\in {\mathbb {N}}\) .
Next, let
\(Be({\varphi })(x):={\varphi }^{\prime \prime }(x)+((2\nu +1)/x){\varphi }^\prime (x)\)  be the Bessel operator
and let
\(D=(Be)^N\left|_{{\mathbb {R}}^m}\right.=(Be)^N,\,D_n=(Be)^N\left|_{\Omega _n}\right.=(Be)^N,\,D_n^*=(Be)^N\left|_{\Omega _n^*}\right.=(Be)^N,\,N\in {\mathbb {Z}}^1_+,\,n\in {\mathbb {N}}\) .
Further, let
\({\mathcal {P}}_{n,1,e}:=\lbrace P\in {\mathcal {P}}_{n,1}:P(-x)=P(x),\,x\in {\mathbb {R}}^1\rbrace \) 
be a subset of \({\mathcal {P}}_{n,1}\)  of all even
univariate polynomials with complex coefficients, and let
\(B_n={\mathcal {P}}_{n,1,e}\left|_{\Omega _n}\right.\)  and
\(B_n^*={\mathcal {P}}_{n,1,e}\left|_{\Omega _n^*}\right.,\,n\in {\mathbb {N}}.\)
Inequalities (REF ) were proved by the author
[9]} in the following form:
\(\lim _{n\rightarrow {\infty }}\sup _{P\in {\mathcal {P}}_{2\lfloor n/2 \rfloor ,1,e}\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(P)(0)\vert }{\Vert P\Vert _{L_{p,\mu }([-n,n])}}&\le &\sup _{f\in (B_{1,e}\cap L_{p,\mu }({\mathbb {R}}^1))\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(f)(0)\vert }{\Vert f\Vert _{L_{p,\mu }({\mathbb {R}}^1)}}\nonumber \\&\le & \lim _{n\rightarrow {\infty }}\sup _{P\in {\mathcal {P}}_{2\lfloor n/2 \rfloor ,1,e}\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(P)(0)\vert }{\Vert P\Vert _{L_{p,\mu }([-\tau n,\tau n])}}\) 
for \(\tau \in (0,1)\) .
By letting \(\tau \rightarrow 1-\)  in (REF ),
we arrive at the following asymptotic result
for \(0<p\le {\infty }\) 
(see [9]}):
\(\sup _{f\in (B_{1,e}\cap L_{p,\mu }({\mathbb {R}}^1))\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(f)(0)\vert }{\Vert f\Vert _{L_{p,\mu }({\mathbb {R}}^1)}}= \lim _{n\rightarrow {\infty }} n^{-2N-(2\nu +2)/p}\sup _{P\in {\mathcal {P}}_{2\lfloor n/2 \rfloor ,1,e}\setminus \lbrace 0\rbrace }\frac{\vert (Be)^N(P)(0)\vert }{\Vert P\Vert _{L_{p,\mu }([-1,1])}}.\) 
Verifications of Conditions C3.1)
and C12.1) in weighted spaces are provided similarly
to Example REF  (see [9]} for details).
So (REF ) for \(p\in (0,{\infty }]\)  follows from
Corollary REF  (v).

Remark 7.5 
Special cases of relations (REF ), (REF ),
and (REF )
that are discussed
in Examples
REF , REF , and REF 
and published in
[6]}, [7]}, [8]}, [9]}, [10]}
follow from statements (ii), (v), and (vi) of
Corollary REF .
In other words, the relations presented in
the aforementioned publications can be proved
by verifying the validity of the corresponding
conditions C3.1) (or C3.1.1)) and C12.1).
Despite the fact that these conditions were
not formulated explicitly in
these papers,
the proofs of
special cases of (REF ), (REF ),
and (REF )
were actually based on the verification of
these conditions.
This observation along with the general approach
to sharp constants of approximation theory
developed in [52]}, [53]} were the major
motivations for writing this paper.

Approximation of Individual Elements
(Problem B)
Here, we discuss a special case of Corollary REF  and
three examples.

General result.
We assume that all assumptions of Section REF  are held
in the following special cases.
First,
we assume that \(F^{(2)}_*=F({\mathbb {R}}^m)=F({\mathbb {R}}^m,\mu )\)  is
a \({\beta }_2\) -normed functional space
with the monotone and \({\omega }\) -continuous \({\beta }_2\) -norm
(see Definitions REF  and REF ), \({\beta }_2\in (0,1]\) , and
\(F^{(2)}_{*n}=F({\Omega }_n)\) 
or \(F^{(2)}_{*n}=F({\Omega }_n^*),\,n\in {\mathbb {N}}\) .
Next, we define the restriction operators \(L_n^{(2)}=R({\mathbb {R}}^m,{\Omega }_n^*),\,l_n^{(2)}=R({\mathbb {R}}^m,{\Omega }_n),\,n\in {\mathbb {N}}\) , and
\(l_{n,s}^{(2)}=\left\lbrace \begin{array}{ll}R({\Omega }_n,{\Omega }_s),&n\ge s,\\R({\Omega }_s,{\Omega }_n),&n< s,\end{array}\right.\quad n\in {\mathbb {N}},\quad s\in {\mathbb {N}}.\) 
We also assume that \(B,\,B_n\) , and \(B_n^*\) 
are nontrivial subspaces of \(F({\mathbb {R}}^m),\ F({\Omega }_n)\) , and \(F({\Omega }_n^*)\) ,
respectively, \(n\in {\mathbb {N}}\) .
Next, for \({\varphi }\in F({\mathbb {R}}^m)\)  we set
\({\varphi }_n:=L_n^{(2)}({\varphi })={\varphi }\left|_{\Omega _n^*}\right.\in F(\Omega _n^*)\) 
or
\({\varphi }_n:=l_n^{(2)}({\varphi })={\varphi }\left|_{\Omega _n}\right.\in F(\Omega _n)\) .
In addition, we assume that
\({\varphi }\in F({\mathbb {R}}^m)\setminus B\)  and
\({\varphi }\left|_{\Omega _n}\right.\in F(\Omega _n)\setminus B_n\)  or
\({\varphi }_n\left|_{\Omega _n^*}\right.\in F(\Omega _n^*)\setminus B_n^*\) .
Note that to the end of the example we use the
notation \({\varphi }\)  instead of
\({\varphi }\left|_{\Omega _n}\right.\)  or
\({\varphi }\left|_{\Omega _n^*}\right.\) .
The "norms" are defined by (REF ) as follows.
\(\Vert h\Vert _{F^{(2)}}:=\Vert {\varphi }-h\Vert _{F({\mathbb {R}}^m)},\quad h\in F({\mathbb {R}}^m);\qquad \Vert h_n\Vert _{F^{(2)}_n}:=\Vert {\varphi }-h_n\Vert _{F(\Omega )},\quad h_n\in F(\Omega ),\) 
where \(\Omega =\Omega _n\)  or \(\Omega =\Omega _n^*\) .
Furthermore, for \({\varphi }\in F({\mathbb {R}}^m)\)  we consider the errors of
best approximation
\(E({\varphi },B,F({\mathbb {R}}^m))\) , \(E({\varphi },B_n,F({\Omega }_n))\) , and
\(E({\varphi },B_n^*,F({\Omega }_n^*)),\,n\in {\mathbb {N}}\)  (see (REF )
or (REF )
for the definition);
by the assumptions on \({\varphi }\)  and its restriction to
\(\Omega _n\)  or \(\Omega _n^*,\,n\in {\mathbb {N}}\) ,
all these approximation errors are positive.
Finally, let
\(B_n^\prime \subseteq B_n\setminus \lbrace 0\rbrace \)  be a set
from Condition C9) such that
the following relation holds:
\(E({\varphi },B_n,F({\Omega }_n))=E({\varphi },B_n^\prime ,F({\Omega }_n)).\) 
Below we introduce some conditions on families
\(\left\lbrace B,F({\mathbb {R}}^m)\right\rbrace ,\left\lbrace B_n,F({\Omega }_n)\right\rbrace \) ,
and \(\left\lbrace B_n^*,F({\Omega }_n^*)\right\rbrace ,\linebreak n\in {\mathbb {N}}\) ,
that are needed for the validity of relations (REF )
and/or (REF ) in function spaces.

Condition C3.2*). For every \(f\in B\setminus \lbrace 0\rbrace \) 
there exists a sequence \(f_n^*\in B_n^*,\,n\in {\mathbb {N}}\) ,
such that
\(\lim _{n\rightarrow {\infty }}\left\Vert f-f_n^*\right\Vert _{F({\Omega }_n^*)}=0.\) 
Condition C10.2). (Compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\)  such that
\(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exist a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural
numbers and an element \(f\in B\) 
such that
\(\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-f\right\Vert _{F\left({\Omega }_{n_k}\right)}=0.\) 
Condition C12.2). (Weak compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\) 
such that
\(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exists \(f\in B\)  such that
for any fixed \(s\in {\mathbb {N}}\) ,
there exists a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural numbers
(\(n_k=n_k(s),\,k\in {\mathbb {N}}\) )
such that
\(\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-f\right\Vert _{F\left({\Omega }_{s}\right)}=0,\) 
The following corollary holds true.
Corollary 7.6 
(i) If Condition C3.2*)
is satisfied, then
\(\limsup _{n\rightarrow {\infty }} E\left({\varphi },B_n^*,F({\Omega }_n^*)\right)\le E\left({\varphi },B,F({\mathbb {R}}^m)\right).\) 
(ii) If
Condition C10.2) is satisfied, then
\(\liminf _{n\rightarrow {\infty }} E\left({\varphi },B_n,F({\Omega }_n)\right)\ge E\left({\varphi },B,F({\mathbb {R}}^m)\right).\) 
(iii) If
Condition C12.2) is satisfied,
then (REF ) holds
true.
(iv)
Let
Condition C3.2*) and either C10.2) or C12.2) be satisfied.
Then
\(\limsup _{n\rightarrow {\infty }} E\left({\varphi },B_n^*,F({\Omega }_n^*\right))\le E\left({\varphi },B,F({\mathbb {R}}^m)\right)\le \liminf _{n\rightarrow {\infty }} E\left({\varphi },B_n,F({\Omega }_n\right)).\) 

Note first that in this proof all elements in discussed
Conditions C1) through C13) are replaced without further notice
by the corresponding elements that are used in Conditions
C3.2*), C10.2), C12.2) and relations (REF ),
(REF ), (REF ). For example, in Condition
C3*) \(B_n\)  is replaced by \(B_n^*\) .
By the assumptions about the functional spaces and operators
introduced above, Conditions C4*), C6), C6b), C11), and C11a)
of Section REF  are satisfied. Indeed,
Conditions C6) and C11) are satisfied by the direct
definitions of the operators whose existence is required
in these conditions.
In addition,
Conditions C4*), C6b), and C11a) are satisfied
by Proposition REF .
For example, for every \(h\in F^{(2)}\)  we obtain by
(REF ) and by Proposition REF  (ii)
\(\liminf _{n\rightarrow {\infty }}\left\Vert l_n^{(2)}(h)\right\Vert _{ F^{(2)}_n}&=&\liminf _{n\rightarrow {\infty }}\left\Vert {\varphi }_n-l_n^{(2)}(h)\right\Vert _{ F(\Omega _n)}=\liminf _{n\rightarrow {\infty }}\left\Vert l_n^{(2)}({\varphi }-h)\right\Vert _{ F(\Omega _n)}\nonumber \\&=&\Vert {\varphi }-h\Vert _{ F({\mathbb {R}}^m)}= \Vert h\Vert _{F^{(2)}}.\) 
Therefore, Condition C6b) is satisfied. Similarly we can prove that
Conditions C4*) and C11a) are satisfied (see also Remark REF ).
Next, Conditions C3.2*), C10.2), and C12.2) of this section
are equivalent to Conditions C3*),
equation () of C10), and
equation () of C12) from Section
REF , respectively.
Therefore, statements (i), (ii), and (iii)
of Corollary REF 
follow immediately
from Corollary REF , while statement (iv) is
a direct corollary of (i), (ii), and (iii).
\(\Box \)
Remark 7.7 
Statements (i) and (ii) of Corollary REF  with
slightly different conditions were obtained in
[52]} for normed spaces and
in
[53]}
for \({\beta }_2\) -normed spaces.

Examples. There are numerous publications
[56]}, [57]}, [58]}, [34]}, [60]}, [35]}, [36]}, [63]}, [32]}, [65]}, [52]}, [53]}, [68]}, [69]} about special cases of Corollary
REF  and their modifications;
here, we call them limit theorems of approximation theory
(see also Section  B).
Examples REF , REF , and REF  below
discuss relations (REF )
proved in these papers in various situations.
Example 7.8 
(Limit theorems for multivariate periodic approximations).
Let
\(F({\mathbb {R}}^m)=L_p({\mathbb {R}}^m),\linebreak {\beta }_2=\min \lbrace 1,p\rbrace \) ,
and let \({\varphi }\in L_p({\mathbb {R}}^m)\cap M_{C,N}\) 
be a fixed function\(,\,0<p\le {\infty }\) .
Next, let
\( B=B_V\cap L_p({\mathbb {R}}^m),\,0<p\le {\infty },\,{\Omega }_n={\Omega }_n^*=Q_{{\gamma }(n)}^m,\,B_n=B_N^*={\mathcal {T}}_{n,V}\cap M_{C_9C,N+2}\left|_{Q_{{\gamma }(n)}^m}\right.={\mathcal {T}}_{n,V}\cap M_{C_9C,N+2},\,n\in {\mathbb {N}}.\) 
Here, \(C>0\)  and \(N\ge 0\)  are fixed constants;
\(C_9\)  is any constant such that \(C_9\ge C_{10}>0\) ,
and \(C_{9}\)  and \(C_{10}\)  are independent
of \(n\)  and \({\varphi }\) . In addition,
a number sequence \(\lbrace {\gamma }(n)\rbrace _{n=1}^{\infty }\)  satisfies
the conditions
\(0<{\gamma }(n)\le \pi n,\quad n\in {\mathbb {N}},\qquad \lim _{n\rightarrow {\infty }}{\gamma }(n)={\infty },\qquad \lim _{n\rightarrow {\infty }}({\gamma }(n))^{2+m/p}n^{-1/(m+2)}=0.\) 
Relations (REF ) were proved by the author [52]}, [53]}
in the following form:
\(\lim _{n\rightarrow {\infty }}E\left({\varphi },{\mathcal {T}}_{n,V}\cap M_{C_9C,N+2},L_p\left(Q_{{\gamma }(n)}^m\right)\right)=E\left({\varphi },B_V\cap L_p({\mathbb {R}}^m),L_p({\mathbb {R}}^m)\right).\) 
For \(p\in [1,{\infty }]\)  equality (REF ) was proved
in [52]} in a more general situation;
the case of \(p\in (0,1)\)  was discussed in
[53]}. For \(m=1,\,V=[-{\sigma },{\sigma }]\) , and
\(p={\infty }\) , (REF ) was established by Bernstein
[58]}.
Condition C3.2*) of approximation of entire
functions of exponential type \(V\) 
by multivariate trigonometric polynomials was verified in
[52]} (see also [8]}).
In addition, we choose
\(B_n^\prime =B_n={\mathcal {T}}_{n,V}\cap M_{C_9C,N+2}\) .
Since \({\mathcal {T}}_{n,V}\subseteq B_V\) ,
the compactness property of \(\cup _{n=1}^{\infty }B_n^\prime \) , stated in C12.2),
can be found in [77]} and
[8]}
(a special case is discussed in [26]}).
So (REF ) for \(p\in (0,{\infty }]\)  follows from Corollary
REF  (iv).

Example 7.9 
(Limit theorems for multivariate polynomial approximations).
Let
\(F({\mathbb {R}}^m)=L_p({\mathbb {R}}^m),\,{\beta }_2=\min \lbrace 1,p\rbrace ,\,B=B_V\cap L_p({\mathbb {R}}^m),\,0<p\le {\infty },\,{\Omega }_n=nV^*,\,{\Omega }_n^*=\tau nV^*,\,\tau \in (0,1),\,B_n={\mathcal {P}}_{n,m}\left|_{\Omega _n}\right.={\mathcal {P}}_{n,m},\,B_n^*={\mathcal {P}}_{n,m}\left|_{\Omega _n^*}\right.={\mathcal {P}}_{n,m},\,n\in {\mathbb {N}}.\)
Relations (REF )
were proved in
[32]}. The following result is a corollary of
(REF ):
if \({\varphi }\in L_{\infty }({\mathbb {R}}^m)\) , then
\(&&E\left({\varphi },B_{ V}\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }({\mathbb {R}}^m)\right)\le \liminf _{n\rightarrow {\infty }}E\left({\varphi },{\mathcal {P}}_{n,m},L_{\infty }\left(nV^*\right)\right)\nonumber \\&&\le \limsup _{n\rightarrow {\infty }}E\left({\varphi },{\mathcal {P}}_{n,m},L_{\infty }\left(nV^*\right)\right)\le \lim _{\tau \rightarrow 1^-}E\left({\varphi },B_{\tau V}\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }({\mathbb {R}}^m)\right),\) 
and if \({\varphi }\in L_p({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m),\,p\in (0,{\infty })\) , then
\(\lim _{n\rightarrow {\infty }}E\left({\varphi },{\mathcal {P}}_{n,m},L_p\left(nV^*\right)\right)=E\left({\varphi },B_{V}\cap L_p({\mathbb {R}}^m),L_p({\mathbb {R}}^m)\right).\) 
For \(p\in [1,{\infty }]\)  relations (REF ) and
(REF ) were proved
in [32]};
the case of \(p\in (0,1)\)  was discussed
in a more general and precise form in
[53]}. For \(m=1,\,V=[-{\sigma },{\sigma }]\) ,
and \(V^*=[-1/{\sigma },1/{\sigma }]\) ,
inequalities (REF ) were established by Bernstein
[34]}. Equality (REF )
for \(m=1,\,V=[-{\sigma },{\sigma }],\,V^*=[-1/{\sigma },1/{\sigma }]\) ,
and \(p\in [1,{\infty })\) 
was proved by Raitsin [63]}.
More precise relations than (REF ) were proved in
[65]}.
Condition C3.2*) of approximation of entire
functions of exponential type \(V\) 
by multivariate algebraic polynomials was verified in
[32]}
(see also [10]}).
In addition, we choose
\(B_n^\prime =\left\lbrace f_n\in {\mathcal {P}}_{n,m}:\Vert f_n\Vert _{L_p(nV^*)}\le \left(2^{{\beta }_2}+1\right)^{1/{\beta }_2}\Vert {\varphi }\Vert _{L_p({\mathbb {R}}^m)}\right\rbrace ,\qquad n\in {\mathbb {N}}.\) 
Then
\(E({\varphi },{\mathcal {P}}_{n,m},L_p(nV^*))= E({\varphi },B_n^\prime ,L_p(nV^*)),\qquad n\in {\mathbb {N}}.\) 
Indeed, given \({\varepsilon }\in \left(0,\Vert {\varphi }\Vert _{L_p({\mathbb {R}}^m)}\right)\) 
and \(n\in {\mathbb {N}}\) , let \(f_{n,{\varepsilon }}\in {\mathcal {P}}_{n,m}\)  satisfy the inequalities
\(\Vert {\varphi }\Vert _{L_p({\mathbb {R}}^m)}\ge E\left({\varphi },{\mathcal {P}}_{n,m},L_p\left(nV^*\right)\right)> \Vert {\varphi }-f_{n,{\varepsilon }}\Vert _{L_p\left(nV^*\right)}-{\varepsilon }.\) 
Then using the \({\beta }_2\) -triangle inequality,
we see that \(f_{n,{\varepsilon }}\in B_n^\prime \) ,
and, in addition,
\(E\left({\varphi },{\mathcal {P}}_{n,m},L_p\left(nV^*\right)\right)=\inf _{{\varepsilon }\in \left(0,\Vert {\varphi }\Vert _{L_p({\mathbb {R}}^m)}\right)}\Vert {\varphi }-f_{n,{\varepsilon }}\Vert _{L_p\left(nV^*\right)}\ge E\left({\varphi },B_n^\prime ,L_p\left(nV^*\right)\right).\) 
Therefore, (REF ) holds true.
The compactness property of \(\cup _{n=1}^{\infty }B_n^\prime \) , stated in C12.2),
was verified by Bernstein [34]} for \(m=1\) 
and by the author [10]} for \(m>1\) .
So (REF ) for \(p\in (0,{\infty }]\)  follows from Corollary
REF  (iv).

Example 7.10 
(Limit theorems for univariate polynomial approximations
with exponential
weight).
Let \(Q:{\mathbb {R}}^1\rightarrow [0,{\infty })\)  be an even differentiable
convex function and let \({\beta }_2\) -norms
\(\Vert \cdot \Vert _{L_{p}(W_p,{\mathbb {R}}^1)},\,p\in (0,{\infty }],\,{\beta }_2=\min \lbrace 1,p\rbrace \) ,
be defined by
(REF ) and (REF ).
In addition, let \(a_n=a_n(Q)\)  be the \(n\) th
Mhaskar-Rakhmanov-Saff number defined as the
positive root of the equation
\(n=\frac{2}{\pi }\int _0^1\frac{a_nx\,Q^\prime (a_nx)}{\sqrt{1-x^2}}dx\) 
and let
\(b_n=b_n(Q):=\frac{2}{\pi }\int _0^1\frac{Q^\prime (a_nx)\sqrt{1-x^2}}{x}dx+n/a_n.\) 
Next, let
\(F^{(2)}_*=L_p({\mathbb {R}}^1),\,B=B_{[-{\sigma },{\sigma }]}\cap L_p({\mathbb {R}}^1),\,F^{(2)}_{*n}=L_p\left(W_p(({\sigma }/b_n)\cdot ),{\mathbb {R}}^1\right),\,{\sigma }>0,\,\Omega _n=\Omega _n^*={\mathbb {R}}^1,\,B_n={\mathcal {P}}_{n,1},\,B_n^*={\mathcal {P}}_{\lfloor n/\tau \rfloor ,1},\,\tau \in (0,1),\,n\in {\mathbb {N}},\,p\in (0,{\infty }]\) .
Relations like (REF )
were proved in
[69]}. The following result is a corollary of
these relations: let \(W_p\)  satisfy some additional conditions
(see [69]});
if \({\varphi }\in L_{\infty }({\mathbb {R}}^1)\) , then
\(&&E\left({\varphi },B_{[-{\sigma },{\sigma }]}\cap L_{\infty }({\mathbb {R}}^1),L_{\infty }({\mathbb {R}}^1)\right)\nonumber \\&&\le \liminf _{n\rightarrow {\infty }}E\left({\varphi }((b_n/{\sigma })\cdot ),{\mathcal {P}}_{n,1},L_{\infty }\left(W_{\infty },{\mathbb {R}}^1\right)\right)\nonumber \\&&\le \limsup _{n\rightarrow {\infty }}E\left({\varphi }((b_n/{\sigma })\cdot ),{\mathcal {P}}_{n,1},L_{\infty }\left(W_{\infty },{\mathbb {R}}^1\right)\right)\nonumber \\&&\le \lim _{\tau \rightarrow 1^-}E\left({\varphi },B_{[-\tau {\sigma },\tau {\sigma }]}\cap L_{\infty }({\mathbb {R}}^1),L_{\infty }({\mathbb {R}}^1)\right),\) 
and if \({\varphi }\in L_p({\mathbb {R}}^1),\,p\in (0,{\infty })\) , then
\(\lim _{n\rightarrow {\infty }}(b_n/{\sigma })^{1/p}E\left({\varphi }((b_n/{\sigma })\cdot ),{\mathcal {P}}_{n,1},L_p\left(W_p,{\mathbb {R}}^1\right)\right)=E\left({\varphi },B_{[-{\sigma },{\sigma }] }\cap L_p({\mathbb {R}}^1),L_p({\mathbb {R}}^1)\right).\) 
For \(p\in (0,{\infty })\)  equality (REF ) was proved
in [69]}, while a stronger version
of relations (REF ) for \(p={\infty }\) 
was established in
[69]}. For the Hermite weight
similar results were obtained in [68]}.
Condition C3.2*) of polynomial approximation
with exponential weights of entire
functions of exponential type follows from
[69]}.
The sets \(B_n^\prime ,\,n\in {\mathbb {N}}\) , can be chosen
like in Example REF .
The compactness property of
\(\cup _{n=1}^{\infty }B_n^\prime \) , stated in C12.2),
is based on [69]}
and verified in [69]}.
So (REF ) for \(p\in (0,{\infty }]\)  follows from Corollary
REF  (iv).

Remark 7.11 
Special cases of relations (REF )
that are discussed
in Examples
REF , REF , and REF 
and published in
[34]}, [63]}, [32]}, [53]}, [69]}
follow from statement (iv) of
Corollary REF .
In other words, the relations presented in
the aforementioned publications can be proved
by verifying the validity of the corresponding
conditions C3.2*) or/and C12.2).
Despite the fact that these conditions were
not formulated explicitly in
these papers,
the proofs of
special cases of (REF )
were actually based on the verification of
these conditions.
This observation was the major
motivation for general approach developed in
[52]}, [53]}.

Approximation on Classes of Elements
(Problem C)
Here, we discuss a special case of Corollary REF  and
two examples.

General result.
We assume that all assumptions of Section REF  are held
in the following special cases.
First,
we assume that \(F^{(1)}_*=F({\mathbb {R}}^m)=F({\mathbb {R}}^m,\mu )\)  is
a \({\beta }_1\) -normed functional space
with the monotone and \({\omega }\) -continuous
\({\beta }_1\) -norm
(see Definitions REF  and REF ),
\({\beta }_1\in (0,1]\) , and
\(F^{(1)}_{*n}=F({\Omega }_n)\) 
or \(F^{(1)}_{*n}=F({\Omega }_n^*),\,n\in {\mathbb {N}}\) .
Next, we define the restriction operators \(L_n^{(1)}=R({\mathbb {R}}^m,{\Omega }_n^*),\,l_n^{(1)}=R({\mathbb {R}}^m,{\Omega }_n),\,n\in {\mathbb {N}}\) , and
\(l_{n,s}^{(1)}=\left\lbrace \begin{array}{ll}R({\Omega }_n,{\Omega }_s),&n\ge s,\\R({\Omega }_s,{\Omega }_n),&n< s,\end{array}\right.\quad n\in {\mathbb {N}},\quad s\in {\mathbb {N}}.\) 
We also assume that \(B,\,B_n\) , and \(B_n^*\) 
are nontrivial subspaces of \(F({\mathbb {R}}^m),\,F({\Omega }_n)\) , and \(F({\Omega }_n^*)\) ,
respectively, \(n\in {\mathbb {N}}\) .
In addition, let \(G\)  be a subspace of \(F({\mathbb {R}}^m),\,G_n\)  be a subspace of \(F({\Omega }_n)\) ,
and \(G_n^*\)  be a subspace of \(F({\Omega }_n^*),\,n\in {\mathbb {N}}\) .
Next, for \(f\in B,\,f_n\in B_n\) , and \(f_n^*\in B_n^*\) 
we consider the errors of
best approximation
\(E(f,G,F({\mathbb {R}}^m)),\,E(f_n,G_n,F({\Omega }_n))\) , and
\(E(f_n^*,G_n^*,F({\Omega }_n^*)),\,n\in {\mathbb {N}}\)  (see (REF )
or (REF )
for the definition).
Note that each of these
errors of
best approximation are \({\beta }_1\) -seminorms
on the corresponding spaces.
The "norms" are defined by (REF ) and ()
as follows.
\(\Vert h\Vert _{F^{(1)}}=E(h,G,F({\mathbb {R}}^m)),\quad h\in F({\mathbb {R}}^m),\quad \Vert h_n\Vert _{F^{(1)}_n}=E(h_n,\tilde{G},F(\Omega )),\quad h_n\in F(\Omega ),\) 
where \(\Omega =\Omega _n\)  or \(\Omega =\Omega _n^*\) 
and \(\tilde{G}=G_n\)  or \(\tilde{G}=G_n^*,\,n\in {\mathbb {N}}\) .
Note that \(\tilde{G}\)  in (REF ) can depend on \(h_n\) .
Finally, let
\(B_n^\prime \subseteq B_n\setminus \lbrace 0\rbrace \)  be a set
from Condition C8) such that
the following relation holds:
\(\sup _{f_n\in B_n}E(f_n,G_n,F({\Omega }_n))=\sup _{f_n\in B_n^\prime }E(f_n,G_n,F({\Omega }_n)).\) 
Below we introduce some conditions on families
\(\left\lbrace B,F({\mathbb {R}}^m)\right\rbrace ,\left\lbrace B_n,F({\Omega }_n)\right\rbrace \) ,
and \(\left\lbrace B_n^*,F({\Omega }_n^*)\right\rbrace ,\linebreak n\in {\mathbb {N}}\) ,
that are needed for the validity of relations (REF )
and/or (REF ) in function spaces.

Condition C3.3). For every \(f\in B\setminus \lbrace 0\rbrace \) 
there exists a sequence \(f_n^*\in B_n^*,\,n\in {\mathbb {N}}\) ,
such that
\(\lim _{n\rightarrow {\infty }}\left\Vert f-f_n^*\right\Vert _{F({\Omega }_n^*)}=0.\) 
Condition C4.3). For every \(h\in B\) ,
\(\liminf _{n\rightarrow {\infty }}E\left(h\left|_{\Omega _n^*}\right.,G_n^*,F({\Omega }^*_n)\right)=\liminf _{n\rightarrow {\infty }}E\left(h,G_n^*,F({\Omega }^*_n))\ge E(h,G,F({\mathbb {R}}^m)\right).\) 
Condition C6.3*a). For every \(h\in F({\mathbb {R}}^m)\) ,
\(\limsup _{n\rightarrow {\infty }}E\left(h\left|_{\Omega _n}\right.,G_n,F({\Omega }_n)\right)=\limsup _{n\rightarrow {\infty }}E\left(h,G_n,F({\Omega }_n))\le E(h,G,F({\mathbb {R}}^m)\right).\) 
Remark 7.12 
Condition C4.3) can be verified by using statements
(ii) or (iii) of Corollary REF ,
while Condition C6.3*a) can be verified by using statement
(i) of Corollary REF .

Condition C10.3*a). (Compactness condition)
For any sequence \(\lbrace f_n\rbrace _{n=1}^{\infty }\)  such that
\(f_n\in B_n^\prime ,\,n\in {\mathbb {N}}\) ,
there exist a sequence \(\lbrace n_k\rbrace _{k=1}^{\infty }\)  of natural
numbers and an element \(f\in B\) 
such that
\(\lim _{k\rightarrow {\infty }}\left\Vert f_{n_k}-f\right\Vert _{F\left({\Omega }_{n_k}\right)}=0.\) 
The following corollary holds true.
Corollary 7.13 
(i) If Conditions C3.3) and C4.3)
are satisfied, then
\(\liminf _{n\rightarrow {\infty }} \sup _{f_n\in B_n^*}E\left(f_n,G_n^*,F({\Omega }_n^*)\right)\ge \sup _{f\in B}E\left(f,B,F({\mathbb {R}}^m)\right).\) 
(ii) Let \(\sup _{f_n\in B_n}E\left(f_n,G_n,F({\Omega }_n)\right)\)  be a
finite number for every \(n\in {\mathbb {N}}\) .
If Conditions C6.3*a) and C10.3*a) are
satisfied, then
\(\limsup _{n\rightarrow {\infty }} \sup _{f_n\in B_n}E\left(f_n,G_n,F({\Omega }_n)\right)\le \sup _{f\in B}E\left(f,B,F({\mathbb {R}}^m)\right).\) 
(iii) Let \(\sup _{f_n\in B_n}E\left(f_n,G_n,F({\Omega }_n)\right)\)  be a
finite number for every \(n\in {\mathbb {N}}\) .
If
Conditions C3.3), C4.3), C6.3*a),
and C10.3*a) are satisfied, then
\(\liminf _{n\rightarrow {\infty }} \sup _{f_n\in B_n^*}E\left(f_n,G_n^*,F({\Omega }_n^*)\right)\ge \sup _{f\in B}E\left(f,B,F({\mathbb {R}}^m)\right)\ge \limsup _{n\rightarrow {\infty }} \sup _{f_n\in B_n}E\left(f_n,G_n,F({\Omega }_n)\right).\) 

Note first that in this proof all elements in discussed
Conditions C1) through C13) and C10*a)
are replaced without further notice
by the corresponding elements that are used in Conditions
C3.3), C4.3), C6.3a), C10.3*a) and relations (REF ),
(REF ), (REF ). For example, in Condition
C3) \(B_n\)  is replaced by \(B_n^*\) .
Next, Conditions C6) and C11) of Section REF 
are satisfied by the direct
definitions of the operators whose existence is required
in these conditions.
Further, Conditions C3.3), C4.3), C6.3*a),
and C10.3*a) of this section
are equivalent to Conditions C3), C4), C6*a)
and C10*a) of Section
REF , respectively.
Therefore, statements (i) and (ii)
of Corollary REF 
follow immediately
from Corollary REF , while statement (iii) is
a direct corollary of (i) and (ii).
\(\Box \)
Remark 7.14 Statements (i) and (ii) of Corollary REF  with
slightly different conditions were obtained in
[52]} for normed spaces and
in
[53]}
for \({\beta }_1\) -normed spaces.

Examples. There is a number of publications
[107]}, [108]}, [35]}, [52]}, [53]}, [112]} about special cases of Corollary
REF  and their modifications;
here, we call them
relations between constants of approximation theory
(see also Section  C).
Examples REF  and REF  below
discuss relations like (REF )
proved in these papers in various situations.
Example 7.15 
(Periodic approximation on \(H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },n,m})\) ).
Let
\(F({\mathbb {R}}^m)=L_{\infty }({\mathbb {R}}^m),\,{\beta }_1=1,\,B=H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }(R^m),\,B_n^*=H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m},\,G=B_V\cap L_{\infty }({\mathbb {R}}^m),\,G_n^*={\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},\,{\Omega }_n^*=Q_{\pi n/2}^m,\,n\in {\mathbb {N}}.\)
Let us choose \(M=M(n)=\pi n/2,\,n\in {\mathbb {N}}\) ,
and let \(f\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }(R^m)\) .
Then by Lemma REF , there exists a function
\(f_n^*=\tilde{f}:{\mathbb {R}}^m\rightarrow {\mathbb {R}}^m\) 
such that \(f_n^*=f\)  on \(Q_{\pi n/2}^m\)  and
\(f_n^*\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}\) .
Since \(f-f_n^*=0\)  on \(Q_{\pi n/2}^m\) ,
Condition C3.3) is satisfied.
Next, Condition C4.3) is satisfied as well. Indeed,
let a number sequence \(\lbrace {\gamma }(n)\rbrace _{n=1}^{\infty }\) 
satisfy (REF ).
Then
for all large enough \(n,\,{\gamma }(n)\le \pi n/2\) ,
since by (REF ), \({\gamma }(n)=o(n)\)  as \(n\rightarrow {\infty }\) .
In addition, for any
\(h\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)\) ,
we have \(h\in M_{{\omega }(2),1} \cap L_{\infty }({\mathbb {R}}^m)\) ,
by (REF ).
Then using relation (REF ) for \(p={\infty },\,C={\omega }(2)\) ,
and \(N=1\) ,
we obtain
\(&& \liminf _{n\rightarrow {\infty }}E\left(h,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n/2)}^m\right)\right)\\&&\ge \lim _{n\rightarrow {\infty }}E\left(h,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{{\gamma }(n))}^m\right)\right)\\&& = E\left(h,B_V\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }({\mathbb {R}}^m)\right),\qquad h\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m).\) 
Thus Condition C4.3) is satisfied.
We recall that \(C_9\)  is the
same constant as in Example
REF , i.e., \(C_9\) 
is any constant such that \(C_9\ge C_{10}>0\) ,
and \(C_9\)  and \(C_{10}\)  are independent of \(n\)  and \(h\) .
Therefore using Corollary REF  (i), we obtain
\(&&\liminf _{n\rightarrow {\infty }}\sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&&\ge \sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n/2}^m\right)\right)\nonumber \\&& \ge \sup _{f\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
Furthermore, we prove that for any
\(f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m},\,n\in {\mathbb {N}}\) , and for large enough \(C_9\ge C_{10}\) ,
the following equality holds true:
\(E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n}^m\right)\right)= E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right).\) 
Indeed, let us set
\({\mathcal {T}}_{n,V,1}:=\left\lbrace g_n\in {\mathcal {T}}_{n,V}:E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\ge \Vert f_n-g_n\Vert _{L_{\infty }\left(Q_{\pi n}^m\right)}-1\right\rbrace .\) 
If \(g_n\in {\mathcal {T}}_{n,V,1}\) , then by (REF ),
\(\vert g_n(x)\vert &\le & E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)+1+\vert f_n(x)\vert \nonumber \\&\le & E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)+1+{\omega }(2)(1+\vert x\vert ),\qquad x\in {\mathbb {R}}^m.\) 
Since by Jackson's theorem
(see [26]}),
\(&& \sup _{n\in {\mathbb {N}}}\sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&&=\sup _{n\in {\mathbb {N}}}\sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n(n\cdot ),{\mathcal {T}}_{nV},L_{\infty }\left(Q_{\pi }^m\right)\right)<{\infty },\) 
we see from (REF ) and (REF ) that
\({\mathcal {T}}_{n,V,1}\subseteq M_{C_{11},1}\) , where
\(C_{11}\)  is independent of \(n\) .
Then setting
\(C_9=\max \lbrace C_{10},C_{11}/{\omega }(2)\rbrace \) ,
we obtain
\(&& E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_9{\omega }(2),3},L_{\infty }\left(Q_{\pi n}^m\right)\right)\le E\left(f_n,{\mathcal {T}}_{n,V}\cap M_{C_{11},1},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&& \le E\left(f_n,{\mathcal {T}}_{n,V,1},L_{\infty }\left(Q_{\pi n}^m\right)\right)= E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right).\) 
Thus (REF ) follows from (REF ).
Finally combining (REF ) and (REF ),
we obtain
\(&&\liminf _{n\rightarrow {\infty }}\sup _{f_n\in H_{{\omega }}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&&=\liminf _{n\rightarrow {\infty }}\sup _{f_n\in H_{{\omega },0}({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&& \ge \sup _{f\in H_{{\omega },0}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right)\nonumber \\&& =\sup _{f\in H_{{\omega }}({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
Next, by Lemma REF , for any
\(f_n\in H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}\) ,
\(E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)=E\left(f_n,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right),\qquad n\in {\mathbb {N}}.\) 
Taking into account a trivial inclusion
\(H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}\subseteq H_{\omega }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)\) ,
we obtain from (REF )
\(&&\limsup _{n\rightarrow {\infty }}\sup _{f_n\in H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&& \le \sup _{f\in H_{\omega }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m) ,L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
Combining (REF ) and (REF ), we arrive at
\(&&\lim _{n\rightarrow {\infty }}\sup _{f_n\in H_{\omega }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi n,m}}E\left(f_n,{\mathcal {T}}_{n,V},L_{\infty }\left(Q_{\pi n}^m\right)\right)\nonumber \\&& = \sup _{f\in H_{\omega }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
The corresponding values of the sharp constants
in (REF ) were found only for \(m=1\) 
and for a concave modulus of continuity \({\omega }\) 
(see [114]}, [115]}).
For the Hölder class \(H^{\lambda }({\mathbb {R}}^m)=H_{\omega }({\mathbb {R}}^m)\)  with
\({\omega }(\tau )=\tau ^{\lambda },\,\tau \in (0,{\infty }),\,0<{\lambda }\le 1\) ,
(REF ) immediately implies the relation
\(&&\lim _{n\rightarrow {\infty }} n^{\lambda }\sup _{f\in H^{\lambda }({\mathbb {R}}^m)\cap \tilde{L}_{{\infty },2\pi ,m}}E\left(f,{\mathcal {T}}_{nV},L_{\infty }\left(Q_{\pi }^m\right)\right)\nonumber \\&& = \sup _{f\in H^{\lambda }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)}E\left(f,B_V\cap L_{\infty }({\mathbb {R}}^m),L_{\infty }\left({\mathbb {R}}^m\right)\right).\) 
A more precise result with
\(H^{\lambda }({\mathbb {R}}^m)\cap L_{\infty }({\mathbb {R}}^m)\)  replaced by \(H^{\lambda }({\mathbb {R}}^m)\) 
was obtained in [52]}.
The corresponding univariate result was proved by Bernstein
[107]}. Bernstein [108]} attempted
to extend this result to the class \(W^sH^{\lambda }({\mathbb {R}}^1)\) 
but his proof was incomplete (see also Example REF ).

Example 7.16 
(Periodic approximation on \(W^sH_{\omega }({\mathbb {R}}^1)\) ).

The author [112]} solved
a relatively long-standing problem of finding
the exact value of
\(\sup _{f\in W^sH_{\omega }({\mathbb {R}}^1)}E(f,B_{[-{\sigma },{\sigma }]},L_{\infty }({\mathbb {R}}^1)),\,s\in Z_+^1\) ,
where \({\omega }\)  is a concave modulus of continuity.
The proof is based on a solution of the corresponding
periodic problem (see Korneichuk [120]},
[114]}) and on a limit relation
between periodic and nonperiodic approximations
like (REF ) (see [112]}).
Note that for \(s=0\)  the problem was solved by
Dzyadyk [115]}. More references can be found
in [112]}.
